In music, a 12-tone row is any of the 12! possible orderings of notes in the Western chromatic scale. The musical notes of a 12-tone composition must always arise in the same order, cycling repeatedly through a predetermined "row" of twelve notes. The repetitive structure of 12-tone music lends itself to mathematical study. In 2003, Hunter and von Hippel investigated symmetry in 12-tone rows, using group theory to enumerate equivalence classes of rows under a group of music-theoretic symmetries. They found that highly symmetric rows constitute just 0.13% of the 12! possibilities, and yet these rows arise in 10% of actual compositions. This result provided strong evidence that composers favor symmetric rows, but leaves us wondering about the remaining 90% of compositions. In this talk, we introduce a way to measure the occurrence of short repetitions and symmetries that go undetected in the analysis of Hunter and von Hippel. We present a new hierarchy of symmetry for 12-tone rows, and offer evidence that composers favor symmetric substructures in their work.